On Thu, Jul 28, 2011 at 1:36 AM, Robert Munafo <mrob27@gmail.com> wrote:
The Mandelbrot set is also sometimes defined as (using the wording on the Wikipedia page) "the limit set of a sequence of plane algebraic curves, the Mandelbrot curves, of the general type known as polynomial lemniscates". These curves are:
L1: |c|=2, L2: |c^2+c|=2, L3: |(c^2+c)^2+c|=2,
and so on. They form a set of "contour lines" which are seen in many early computer images, including some in Mandelbrot's 1986 book. They first few are illustrated (with formulas) here: http://en.wikipedia.org/wiki/File:Lemniscates5.png
So my guess at making sense of Mandelbrot's "conjecture" statement is that, the "limit of lemniscates" definition might imply MLC. In other words, if it can be shown that the lemniscates L1, L2, L3, ... converge on the boundary of the Mandelbrot set, then perhaps such a proof would also prove MLC. If so, then the MLC conjecture is equivalent to a conjecture that the two ways of drawing the Mandelbrot set (pixel scanning and tracing lemniscates) yield the same image, and Mandelbrot's statements would make sense.
Except that the statement that the leminscates converge to the boundary of the mandelbrot set is easily seen to be true. If a point is outside of Ln, then one of the iterates has norm > 2, and further iteration will increase the norm, so the point is not in the Mandelbrot set. And if a point is inside Ln for every n, then all the iterates lie within L1, so the iterates don't diverge, and the point is in the mandelbrot set. Here is a half-remembered, and possibly mangled, conjecture that two definitions both define the Mandelbrot set. Let f_c be the map f_c(z) = z^2 + c. And let f_c^n be the nth iterate of f_c. Among the points in the Mandelbrot set, we have at least the following points: A. Fixed points of iterates; Those points c for which f_c^n(0) = 0 for some n. B. Points in the basin of attraction of fixed points of iterates: Suppose there exist z and n for which f_c^n(z) = z, and |f_c^n'(z)| < 1. Then for points w sufficiently near z, the sequence f_c^(n*m)(w) converges to z. If 0 is sufficiently near z, then c is in the Mandelbrot set. More succinctly, these are the values c for which there exists an n such that lim_(m->infinity) f_c^nm(0) exists. C. The Mandelbrot set is closed, (it is the intersection of the closed sets |f_c^n(0)| < 100), so any set in the closure of A and B is in the Mandelbrot set. There could be another category I'm leaving out, but I think the unproved conjecture is that this is all there is, that is, that every point in the Mandelbrot set is in one of these 3 categories. But I don't see a good way to plot the points in category B, so I can't reconcile this with Mandelbrot's claim that the conjecture seemed to be true experimentally. Andy